Vol. 66, No. 2, 1976

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Spaces with bases satisfying certain order and intersection properties

William Lindgren and Peter Joseph Nyikos

Vol. 66 (1976), No. 2, 455–476

An ortho-base is a base such that the intersection of any subcollection is either an open set or a singleton for which the subcollection is a local base. This paper is primarily devoted to the relationship of bases of this sort to other topological properties of bases, and of the space itself. In §1, we compare ortho-bases to bases having similar properties, and study the relationship with developability and paracompactness. In §2, we define “rank” of a base for arbitrary cardinals and show how it is related to orthocompactness, ortho-bases, and bases of countable order. Section 3 treats two related ascending chain conditions in relation to bases of sub-infinite rank and ortho-bases. Section 4 relates the possession of an ortho-base to a number of “generalized metrlc” properties such as first countability, quasi-developability, and quasi-metrizability. The remaining two sections give examples illustrating the various properties and raise a number of unsolved problems.

Mathematical Subject Classification 2000
Primary: 54D20
Received: 1 July 1975
Published: 1 October 1976
William Lindgren
Peter Joseph Nyikos